Do you consider the term jarhead and other names for the different branches of service offensive?

Not a chance. Each service gives each other a hard time, its like siblings picking on each other. However we all pick on the Coast Guard (just kidding)

All depends on how they used the term and the manner how it was used.

I trained with Marines and we had Marines attached to my Battalion as Combat advisors. So I treat them with respect.

I'm not a Marine so the term "jarhead" which I've used with my nephews is not offensive. I served in the Army but not the infantry so "grunt" is inappropriate for me. I do say that I served in the oxymoron "Army Intelligence CIC".
There is one word that absolutely disgusts me and it's the "c" word for the covers which are worn. I'm not a prude. However, that word turns me off then and more so now.

Actually no. there is such a thing as military rivalry and in my days in the military it was taken very well. Weather you are a Marine, Air Force, Navy or Army you have a kinship type of respect for other Branches. While you always believe your Branch to be the best you know that in times of conflict your very happy to see the Marines, Navy and Air Force in the fight with you!

Nope I wear the term Jarhead, or Devil Dog proudly and will till the day I die, I earned that title.

Absolutely not! I'm in the American Legion and we have all branches and ages and not a single one hasn't joked about another's service. Airmen here but one of the older guys who was Navy (my son is also Navy) loves to pick on the ARMY which he says means {Ain't Really Marines Yet} :-). We are family, after all!

I am sure that it is no longer acceptable to run past the Navy barracks at zero dark thirty singing "Navy Navy don't be blue, Sinbad was a F@GG@T too. However, inter service rivalry is an excellent way to build Esprit de Corps.

I never took offense to it. To me it was a badge of honor.

Being in the Seabees and serving with them some times I'd refer to them as Uncle Sam's Misguided Children or what the definition of a Marine was......A Seabee on light Duty